How to stop sap from seeping out

I think there's something in between getting the stain out and "refinishing" the table, and that's applying a food-safe oil finish to the entire table. You could wipe on (and back off) some linseed oil, tung oil, or even more salad dressing for that matter. This would be a lot less effort than any kind of refinishing and should do a lot to make the stains look less obvious.

Note that I'm assuming that, since the wood table has no current finish it's not a museum piece or anything and is meant to look a little rustic.

If its hollow then there no way to fix it unless its just on the surface. If its painted than you can just apply some wood filler and sand smooth and repaint the area. It will look as good as new.